Power flow control apparatus

ABSTRACT

A power flow control apparatus comprising a current distribution circuit arranged to distribute an input current into a plurality of branches such that the input current is distributed into a plurality of individual branch currents; wherein each of the plurality of branches includes an inductive arrangement arranged to form an inductive coupling with an associated inductive arrangement of at least one other associated branch, and a plurality of compensator units in electrical communication with the plurality of branches, wherein each compensator unit is arranged to deliver a branch compensating voltage relative to the branch current.

TECHNICAL FIELD

The present invention relates to a power flow control apparatus and particularly, although not exclusively, to a compensator circuit for regulating power flow in a two machine system.

BACKGROUND

Various power flow control techniques for regulating power flow and transfer over a transmission line have been reported in the literature.

For example, a flexible alternating current transmission system (FACTS) can be used for the AC transmission of electrical energy. FACTS are generally power electronics based systems that can enhance controllability and increase power transfer capability of a network. In some applications, they can also improve power oscillation damping on power grids.

However, simple FACTS systems often fail to meet the required power transfer capacity for wide ratings diversity of the local generators and loads in distributed generation systems. Thus, a more flexible means of managing a wide range of power transfer requirements is desired.

SUMMARY OF THE INVENTION

In accordance with a first aspect of the present invention, there is provided a power flow control apparatus comprising a current distribution circuit arranged to distribute an input current into a plurality of branches such that the input current is distributed into a plurality of individual branch currents; wherein each of the plurality of branches includes an inductive arrangement arranged to form an inductive coupling with an associated inductive arrangement of at least one other associated branch; and a plurality of compensator units in electrical communication with the plurality of branches, wherein each compensator unit is arranged to deliver a branch compensating voltage relative to the branch current.

In an embodiment of the first aspect, the compensator units are disposed downstream of the current distribution circuit.

In an embodiment of the first aspect, each compensator unit is arranged to regulate its branch compensating voltage.

In an embodiment of the first aspect, each compensator unit is further arranged to automatically regulate its branch compensating voltage in response to a command.

In an embodiment of the first aspect, the power flow control apparatus is further arranged to provide an output voltage based on the branch compensating voltages of the compensator units.

In an embodiment of the first aspect, the input current is a current on a transmission line having at least one current source.

In an embodiment of the first aspect, the current distribution circuit distributes the input current into the plurality of individual branch currents based on at least one predetermined ratio of the current distribution circuit.

In an embodiment of the first aspect, the inductive arrangement of each branch of the current distribution circuit comprises a first coil and a second coil.

In an embodiment of the first aspect, the at least one predetermined ratio of the current distribution circuit is associated with a number of turns of the first coil and a number of turns of the second coil.

In an embodiment of the first aspect, the first coil of each branch is inductively coupled with the second coil of an adjacent branch.

In an embodiment of the first aspect, the first coil of each branch and the second coil of an adjacent branch together define a transformer unit.

In an embodiment of the first aspect, the first coils are primary coils of the transformer units and the second coils are secondary coils of the transformer units.

In an embodiment of the first aspect, the primary coil and the secondary coil of each transformer unit of the current distribution circuit are disposed in adjacent branches.

In an embodiment of the first aspect, the primary coil of each transformer unit of the current distribution circuit is disposed on the same branch with the secondary coil of an adjacent transformer unit of the current distribution circuit.

In an embodiment of the first aspect, the primary coils each comprises a first number of turns of coils and the secondary coils each comprises a second number of turns of coils.

In an embodiment of the first aspect, each transformer unit of the current distribution circuit has a turn ratio defined by the first number of turns of coils of the primary coil and the second number of turns of coils of the secondary coil.

In an embodiment of the first aspect, the at least one predetermined ratio of the current distribution circuit is determined by at least one of the turn ratio of at least one of the transformer units.

In an embodiment of the first aspect, a summation of the individual branch current of each of the plurality of individual branches of the current distribution circuit is substantially equal to the input current.

In an embodiment of the first aspect, the compensator units are static synchronous series compensators (SSSC) each comprising a voltage source converter.

In an embodiment of the first aspect, each static synchronous series compensator is arranged to provide the branch compensating voltage.

In an embodiment of the first aspect, each static synchronous series compensator is further arranged to regulate its branch compensating voltage.

In an embodiment of the first aspect, each static synchronous series compensator is further arranged to automatically regulate its branch compensating voltage in response to a command.

In an embodiment of the first aspect, the power flow control apparatus is arranged to provide an output voltage based on the branch compensating voltages of the static synchronous series compensators.

In an embodiment of the first aspect, the number of transformer units equals the number of branches.

In an embodiment of the first aspect, the number of compensator units equals the number of branches.

In an embodiment of the first aspect, the number of compensator units equals the number of transformer units.

In an embodiment of the first aspect, the transformer units are connected in a daisy-chained manner.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the present invention will now be described, by way of example, with reference to the accompanying drawings in which:

FIG. 1 is a diagram illustrating a simplified current distribution scheme;

FIG. 2 is a diagram illustrating a current distribution circuit of a power flow control apparatus in accordance with one embodiment of the present invention;

FIG. 3 is a diagram illustrating a branch of the current distribution circuit of FIG. 2;

FIG. 4 is a diagram showing a power flow control apparatus in accordance with one embodiment of the present invention being disposed in an elementary two-machine system;

FIG. 5 is a diagram of a compensator unit of the power flow control apparatus of FIG. 4;

FIG. 6A is a diagram of a steady state equivalent circuit of the compensator unit of FIG. 5;

FIG. 6B is diagram of an a.c. small signal equivalent circuit of the compensator unit of FIG. 5;

FIG. 7 is a small signal control block diagram of the power flow control apparatus of FIG. 4;

FIG. 8A is a plot illustrating the relationship of the maximum overshoot M_(p) against the integral gain K_(i) and the proportional gain K_(p) of the power flow control apparatus of FIG. 4 under different values of current I_(r).

FIG. 8B is a plot illustrating the relationship of the settling time t_(s) against the integral gain K_(i) and the proportional gain K_(p) of the power flow control apparatus of FIG. 4 under different values of current

FIG. 9 is a diagram showing the steady state voltage and current waveforms of the compensator units in each branch when a power of 965 W is transferred from v_(A) to v_(B) through a power flow control apparatus in accordance with one embodiment of the present invention.

FIG. 10 is a diagram showing the steady state voltage and current waveforms of the compensator units in each branch when a power of 595 W is transferred from v_(B) to v_(A) through the power flow control apparatus of FIG. 9.

FIG. 11 is a diagram showing the transient responses of the current and voltage of the compensator units in each branch when the power flow control apparatus of FIG. 9 is changed from the operation of FIG. 9 to the operation of FIG. 10.

DETAILED DESCRIPTION OF THE INVENTION

With reference to FIG. 4, there is illustrated an embodiment of a power flow control apparatus comprising a current distribution circuit arranged to distribute an input current into a plurality of branches such that the input current is distributed into a plurality of individual branch currents; wherein each of the plurality of branches includes an inductive arrangement arranged to form an inductive coupling with an associated inductive arrangement of at least one other associated branch; and a plurality of compensator units in electrical communication with the plurality of branches, wherein each compensator unit is arranged to deliver a branch compensating voltage relative to the branch current.

In the following description, an embodiment of the current distribution circuit in the power flow control apparatus of the present invention will first be described. Then, the modelling, design and analysis of the power flow control apparatus in one embodiment of the present invention will be provided. Lastly, experimental results of the power flow control apparatus will be presented.

Referring now to FIG. 1, there is shown a diagram illustrating a simplified current distribution scheme 100. The inventors through their research, trials and experimentation have devised that an alternating electric current can be divided into a plurality of branches based a plurality of predefined parameters. For example, as shown in FIG. 1, a main current i_(M) is shared among N branches. The currents in the branches are denoted as i₁, i₂, . . . , i_(k), . . . , i_(N). More specifically, the main current is related to the branch currents by the following equation:

i ₁ +i ₂ + . . . +i _(k) + . . . +i _(M) =i _(M)  (1).

Preferably, the currents are alternating current (AC). In some other embodiments, however, the currents may also be direct currents (DC) in another embodiment.

With reference to FIG. 2, there is illustrated a current distribution circuit 200 of a power flow control apparatus in accordance with one embodiment of the present invention. In this embodiment, the current distribution circuit 200 is constructed by a number of magnetically coupled inductive components (transformers or transformer units) 202 connected in a daisy chained manner 208. As shown in FIG. 2, the primary coil 206 and the secondary coil 204 of each transformer unit 202 are disposed in adjacent branches. In particular, the primary coil 206 of each transformer unit 202 is disposed in the same branch as the secondary coil 204 of an adjacent transformer unit 202. Preferably, the number of the transformers 202 required is equal to the number of branches.

In this embodiment, the current flowing through each branch is determined by the turns ratios (ratio of the number of turns of coils in the primary coil 206 to the number of turns of coils in the secondary coil 204) of the transformers 202, i.e. n₁, n₂, . . . , n_(k), . . . , n_(N). Ideally, all transformers 202 have infinite magnetizing inductances. Accordingly, the currents in the branches can be expressed as follows:

$\begin{matrix} {i_{1} = {n_{1}i_{N}}} & (2) \\ {{i_{2} = {n_{2}i_{1}}}\vdots} & (3) \\ {{i_{k} = {n_{k}i_{k - 1}}}\vdots} & (4) \\ {i_{N} = {n_{N}{i_{N - 1}.}}} & (5) \end{matrix}$

Thus, by substituting equations (2) to (5) into equation (1), it can be shown that

$\begin{matrix} {i_{k} = {\frac{\prod\limits_{j = 1}^{k}\; n_{j}}{n_{1} + {n_{1}n_{2}} + \ldots + {n_{1}n_{2}\mspace{14mu} \ldots \mspace{14mu} n_{k}} + \ldots + {n_{1}n_{2}\mspace{14mu} \ldots \mspace{14mu} n_{N}}}{i_{M}.}}} & (6) \end{matrix}$

In other words, equation (6) shows that the current in any one of the branches will depend on the turn ratios of the transformers 202 of the current distribution circuit 200. In particular, an advantage of this embodiment is that the current division is substantially independent of the branch voltages v₁, v₂, . . . , v_(k), . . . , v_(N), and v_(M).

Furthermore, in one particular embodiment, when n₁=n₂= . . . =n_(k)= . . . =n_(N)=1,

$\begin{matrix} {i_{1} = {i_{2} = {\ldots = {i_{k} = {\ldots = {i_{N} = {\frac{1}{N}{i_{M}.}}}}}}}} & (7) \end{matrix}$

This is advantageous in that the current i_(M) is equally shared by the branches.

With reference to FIG. 3, there is shown a branch 300 of the current distribution circuit of FIG. 2. In reality, practical transformers may have finite magnetizing inductance, leakage inductance, and resistance. FIG. 3 shows the equivalent circuit of the k-th branch, in which L_(m,k) is the magnetizing inductance of the transformer T_(k) and L_(k) is the equivalent series inductance of the branch. Preferably, L_(k) includes the leakage inductance of the transformer. In some embodiments, the resistance can be neglected.

By applying the Kirchhoff's voltage law to the branch 300, it can be shown that

v _(M) +v _(T,k) −v _(L,k) −n _(k+1) v _(T,k+1) −v _(k)=0  (8)

in which v_(T,k) is the voltage across the magnetizing inductance L_(m,k) of the transformer 71, v_(L,k) is the voltage across the equivalent series inductance L_(k) of the branch 300, v_(M) is the voltage at the input, n_(k) is the turn ratio of the transformer T_(k) and v_(k) is the voltage at the output of the branch 300.

Accordingly, by using equation (8) for the N branches, it can be shown that the voltages v_(T,k), v_(L,k), v_(M), and v_(k) are related to the turn ratio n_(k) of the transformer T_(k) by the following:

$\begin{matrix} {\begin{bmatrix} {- 1} & n_{2} & 0 & 0 & \ldots & 0 & 0 \\ 0 & {- 1} & n_{3} & 0 & 0 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & 0 & \ldots & {- 1} & n_{k} & \ldots & 0 \\ \vdots & \vdots & \ddots & \ddots & \ddots & \ddots & \vdots \\ 0 & 0 & \ldots & 0 & \ldots & {- 1} & n_{N} \\ n_{1} & 0 & \ldots & 0 & \ldots & 0 & {- 1} \end{bmatrix}{\quad{\begin{bmatrix} v_{T,1} \\ v_{T,2} \\ \vdots \\ v_{T,k} \\ \vdots \\ v_{T,{N - 1}} \\ v_{T,N} \end{bmatrix} = {\begin{bmatrix} {v_{M} - v_{1}} \\ {v_{M} - v_{2}} \\ \vdots \\ {v_{M} - v_{k}} \\ \vdots \\ {v_{M} - v_{N - 1}} \\ {v_{M} - v_{N}} \end{bmatrix} - \begin{bmatrix} v_{L,1} \\ v_{L,2} \\ \vdots \\ v_{L,k} \\ \vdots \\ v_{L,{N - 1}} \\ v_{L,N} \end{bmatrix}}}}} & (9) \end{matrix}$

On the other hand, in this embodiment, the voltage V_(L,k) across the equivalent series inductance L_(k) of the k-th branch can be expressed as:

v _(L,k) =sL _(k) i _(k)  (10)

where s=jω is the Laplace operator and w is the operating frequency.

By using equation (10) for N branches, it can be shown that

$\begin{matrix} {\begin{bmatrix} v_{L,1} \\ v_{L,2} \\ \vdots \\ v_{L,k} \\ \vdots \\ v_{L,{N - 1}} \\ v_{L,N} \end{bmatrix} = {\begin{bmatrix} {sL}_{1} & 0 & 0 & 0 & \ldots & 0 & 0 \\ 0 & {sL}_{2} & 0 & 0 & 0 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & 0 & \ldots & {sL}_{k} & 0 & \ldots & 0 \\ \vdots & \vdots & \ddots & \ddots & \ddots & \ddots & \vdots \\ 0 & 0 & \ldots & 0 & \ldots & {sL}_{N - 1} & 0 \\ 0 & 0 & \ldots & 0 & \ldots & 0 & {sL}_{N} \end{bmatrix}{\quad{\begin{bmatrix} i_{1} \\ i_{2} \\ \vdots \\ i_{k} \\ \vdots \\ i_{N - 1} \\ i_{N} \end{bmatrix}.}}}} & (11) \end{matrix}$

Equation (11) illustrates that the voltage v_(L,k) across the equivalent series inductance L_(k) of the k-th branch is related to the current i_(k) of the k-th branch.

In this embodiment, as illustrated above, the current among difference branches are inter-related. Furthermore, by applying the Kirchhoff's current law, it can be shown that

$\begin{matrix} {{{{- n_{k}}i_{k - 1}} + i_{k}} = {- \frac{v_{T,k}}{s\; L_{m,k}}}} & (12) \end{matrix}$

in which n_(k) is the turn ratio of the transformer T_(k), i_(k) is the current in the k-th branch, L_(m,k) is magnetizing inductance of the transformer T_(k), v_(T,k) is the voltage across the magnetizing inductance L_(m,k) of the transformer T_(k) and s=jω is the Laplace operator where ω is the operating frequency. Preferably, in the above expression, when k=1, (k−1)=N.

By arranging equation (12) into matrix form, the following equation can be obtained:

$\begin{matrix} {\begin{bmatrix} 1 & 0 & 0 & 0 & \ldots & 0 & {- n_{1}} \\ {- n_{2}} & 1 & 0 & 0 & 0 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & 0 & \ldots & {- n_{k}} & 1 & \ldots & 0 \\ \vdots & \vdots & \ddots & \ddots & \ddots & \ddots & \vdots \\ 0 & 0 & \ldots & 0 & \ldots & 1 & 0 \\ 0 & 0 & \ldots & 0 & \ldots & {- n_{N}} & 1 \end{bmatrix}{\quad{\begin{bmatrix} i_{1} \\ i_{2} \\ \vdots \\ i_{k} \\ \vdots \\ i_{N - 1} \\ i_{N} \end{bmatrix} = {\quad{\begin{bmatrix} {- \frac{1}{{sL}_{m,1}}} & 0 & 0 & 0 & \ldots & 0 & 0 \\ 0 & {- \frac{1}{{sL}_{m,2}}} & 0 & 0 & 0 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & 0 & \ldots & {- \frac{1}{{sL}_{m,k}}} & 0 & \ldots & 0 \\ \vdots & \vdots & \ddots & \ddots & \ddots & \ddots & \vdots \\ 0 & 0 & \ldots & 0 & \ldots & {- \frac{1}{{sL}_{m,{N - 1}}}} & 0 \\ 0 & 0 & \ldots & 0 & \ldots & 0 & {- \frac{1}{{sL}_{m,N}}} \end{bmatrix}{\quad\begin{bmatrix} v_{T,1} \\ v_{T,2} \\ \vdots \\ v_{T,k} \\ \vdots \\ v_{T,{N - 1}} \\ v_{T,N} \end{bmatrix}}}}}}} & (13) \end{matrix}$

As shown in equation (13), the voltage v_(T,k) across the magnetizing inductance of the transformer T_(k) is related to the current 4 in the k-th branch.

In this embodiment, by further using equations (9), (11) and (13) (eliminating v_(T,k) and v_(L,k) from equation (9)), it can be shown that

$\begin{matrix} {\begin{bmatrix} i_{1} \\ i_{2} \\ \vdots \\ i_{k} \\ \vdots \\ i_{N - 1} \\ i_{N} \end{bmatrix} = {\lbrack Y\rbrack \begin{bmatrix} {v_{M} - v_{1}} \\ {v_{M} - v_{2}} \\ \vdots \\ {v_{M} - v_{k}} \\ \vdots \\ {v_{M} - v_{N - 1}} \\ {v_{M} - v_{N}} \end{bmatrix}}} & (14) \end{matrix}$

where [Y]=[Z]⁻¹ and

$\lbrack Z\rbrack = \begin{bmatrix} \begin{matrix} {{sL}_{m,1} + {n_{2}^{2}{sL}_{m,2}} +} \\ {sL}_{1} \end{matrix} & {{- n_{2}}{sL}_{m,2}} & 0 & 0 & \ldots & 0 & {{- n_{1}}{sL}_{m,1}} \\ {{- n_{2}}{sL}_{m,2}} & \begin{matrix} {{sL}_{m,2} + {n_{3}^{2}{sL}_{m,3}} +} \\ {sL}_{2} \end{matrix} & {{- n_{3}}{sL}_{m,3}} & 0 & 0 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & 0 & \ldots & {{- n_{k}}{sL}_{m,k}} & \begin{matrix} {{sL}_{m,k} +} \\ {{n_{k + 1}^{2}{sL}_{m,{k + 1}}} +} \\ {sL}_{k} \end{matrix} & {{- n_{k + 1}}{sL}_{m,{k + 1}}} & 0 \\ \vdots & \vdots & \ddots & \ddots & \ddots & \ddots & \vdots \\ 0 & 0 & \ldots & 0 & \ldots & \begin{matrix} {{sL}_{m,{N - 1}} +} \\ {{n_{N}^{2}{sL}_{m,N}} +} \\ {sL}_{N - 1} \end{matrix} & 0 \\ {{- n_{1}}{sL}_{m,1}} & 0 & \ldots & 0 & \ldots & {{- n_{N}}{sL}_{m,N}} & \begin{matrix} {{sL}_{m,N} + {n_{1}^{2}{sL}_{m,1}} +} \\ {sL}_{N} \end{matrix} \end{bmatrix}$

Equation (14) shows the relationship of the current i_(k) in the k-th branch and the voltage difference v_(M)−v_(k) across the respective branch. In a preferred embodiment, [Z] is arranged such that the current i_(k) of the k-th branch is substantially independent of the voltage difference v_(m)−v_(k) across the respective branch.

A further illustration of the current distribution circuit in one embodiment is provided as follows. In the following example, the current distribution circuit is arranged to have three branches. In the following, v_(M), v₁, v₂, and v₃ are dc voltages. By arranging the magnetization inductance in each branch to be the same (L_(m,1)=L_(m,2)=L_(m,3)=L_(m)), and the equivalent series inductance in each branch to be the same (L₁=L₂=L₃=L), it can be shown that

$\begin{matrix} {{i_{1}(t)} = {\frac{{L\left( {v_{M} - v_{1}} \right)} + {L_{M}\left( {{3v_{M}} - v_{1} - v_{2} - v_{3}} \right)}}{L\left( {L + {3L_{m}}} \right)}t}} & (15) \\ {{i_{2}(t)} = {\frac{{L\left( {v_{M} - v_{2}} \right)} + {L_{M}\left( {{3v_{M}} - v_{1} - v_{2} - v_{3}} \right)}}{L\left( {L + {3L_{m}}} \right)}t}} & (16) \\ {{i_{3}(t)} = {\frac{{L\left( {v_{M} - v_{3}} \right)} + {L_{M}\left( {{3v_{M}} - v_{1} - v_{2} - v_{3}} \right)}}{L\left( {L + {3L_{m}}} \right)}t}} & (17) \end{matrix}$

Advantageously, in this embodiment, it can be observed that if L_(m) is sufficiently large, the currents i(t) of the three branches will be substantially the same.

Although in the above example, the current distribution circuit is arranged to have three branches. However, in some other embodiments, the current distribution circuit may have any number of branches and the current in each branch may not necessarily have to be equal.

Referring now to FIG. 4, there shows an elementary two-machine system 400 with two voltage sources v_(A) and v_(B) interconnected by a lossy transmission line with resistance R and inductance L (with reactance X_(L)). A power flow control apparatus 402 in one embodiment of the present invention is connected in series with the transmission line. Preferably, the power flow control apparatus 402 comprises a current distribution circuit 404 arranged to distribute a transmission line current into a plurality of branches such that the transmission line current is distributed into a plurality of individual branch currents; wherein each of the plurality of branches includes an inductive arrangement arranged to form an inductive coupling with an associated inductive arrangement of at least one other associated branch; and a plurality of compensator units (or compensators) 406 in electrical communication with the plurality of branches, wherein each compensator unit is arranged to deliver a branch compensating voltage relative to the branch current. In particular, in a preferred embodiment, the compensator units 406 are static synchronous series compensators (SSSC) which comprises a voltage source inverter whereas the power flow control apparatus 402 comprises a multi-parallel connected static synchronous series compensators (MSSSC) architecture. Preferably, each branch has one compensator unit 406. However, in other embodiments, each branch can have more than one compensator unit 406 or some of the branches may not have any compensator units 406.

In a preferred embodiment, the power flow P between the two voltage sources v_(A) and v_(B) can be adjusted by controlling the output voltage v_(q) of the power flow control apparatus 402. The transmission line current is shared among the compensator units through daisy-chained transformers T₁, T₂, . . . , T_(N). In one embodiment, the primary and secondary sides of each transformer are connected to two compensators 406. Preferably, the currents through the two connected compensators 406 are in a ratio determined by the transformer turns-ratio. In one embodiment, the number of transformers equals the number of branches.

The modelling of the two-machine system 400 with the power flow control apparatus 402 is provided as follows.

With reference to FIG. 4, the compensator current i=[i₁ i₂ . . . i_(N)]^(T) can be expressed in terms of v_(A), v_(B) and the compensator voltages v=[v₁ v₂ . . . v_(N)]^(T) as

i=G[β(v _(A) −v _(B))−v]  (18)

where β=[1 1 . . . 1 1]^(T) and G={KZ+(sL+R)[1]}⁻¹ in which

$\mspace{20mu} {K = {\begin{bmatrix} 1 & 0 & \ldots & 0 & {- n_{N}} \\ {- n_{1}} & 1 & 0 & \ldots & 0 \\ 0 & {- n_{2}} & 1 & 0 & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ 0 & \ldots & 0 & {- n_{N - 1}} & 1 \end{bmatrix}{\quad{,{Y = {\begin{bmatrix} {sL}_{m,1} & {{- n_{1}}{sL}_{m,1}} & 0 & \ldots & 0 \\ 0 & {sL}_{m,2} & {{- n_{2}}{sL}_{m,2}} & \ldots & \vdots \\ \vdots & \vdots & \ddots & \ddots & 0 \\ 0 & 0 & \ldots & \ddots & {{- n_{N - 1}}{sL}_{m,{N - 1}}} \\ {{- n_{N}}{sL}_{m,N}} & 0 & \ldots & 0 & {sL}_{m,N} \end{bmatrix}{\quad{,L_{m,k}}}}}}}}}$

is magnetizing inductance of the transformer T_(k), n_(k) is the turn ratio of the transformer T_(k), s=jω is the Laplace operator with ω being the operating frequency, L is the inductance of the transmission line, R is the resistance of the transmission line and [1] is N×N unity matrix.

Moreover, a describing function showing the small-signal response of the compensator output (compensating) currents to the variations in the compensator output (compensating) voltages is given by

Δi(s)=−GΔv(s)  (19)

where Δi(s) and Δv(s) are the small-signal variations in i and v respectively.

The modelling of a compensator unit 406 of the power flow control apparatus 402 is provided below.

FIG. 5 shows the circuit schematic 500 of the r-th compensator unit 406 in FIG. 4. FIGS. 6A and 6B show respectively a steady-state low-frequency model 600 and an a.c. small-signal model 602 of the compensator unit 406. With reference to FIG. 6A, it can be shown that

$\begin{matrix} {{v_{r}(t)} = {{L_{q,r}\frac{{i_{r}(t)}}{t}} + {m\; {v_{{dc},r}(t)}\; {\sin \left\lbrack {{\omega \; t} + {\theta_{r}(t)}} \right\rbrack}}}} & (20) \\ {{i_{{dc},r}(t)} = {\frac{1}{2}m\; {i_{r}(t)}\; {\sin \left\lbrack {{\omega \; t} + {\theta_{r}(t)}} \right\rbrack}}} & (21) \end{matrix}$

where v_(dc,r)(t) is voltage across the dc capacitor C_(dc,r).

By injecting small-signal perturbations into v_(r)(t), i_(r)(t), v_(dc,r)(t), and θ(t) in equations (20) and (21), it can be shown that

$\begin{matrix} {{\Delta \; {v_{r}(t)}} = {{\frac{m^{2}I_{r}}{4\omega_{e}C_{{dc},r}}{\Delta\Theta}_{r}} + {\sin\left( {{\omega_{e}t} - {90{^\circ}}} \right\rbrack}}} & (22) \\ {{\Delta \; {i_{r}(t)}} = {\frac{1}{2}\Delta \; I_{r}\sin \; \omega_{i}\; t}} & (23) \end{matrix}$

where ΔI_(r) and ω_(i) are the amplitude and frequency of perturbation in i_(r)(t), ΔΘ, and ω_(e) are the amplitude and frequency of perturbation in θ_(r)(t).

In addition, two describing functions, D_(i)(s) and D_(θ)(s) for studying the variations of v_(r)(t) with respect to i_(r)(t) and θ_(r)(t) are derived as

$\begin{matrix} {{{D_{i}(s)} = {\left. \frac{\Delta \; {v_{r}(s)}}{\Delta \; {i_{r}(s)}} \right|_{{\Delta \; \theta_{r}} = 0} = 0}},{{D_{\theta}(s)} = {\left. \frac{\Delta \; {v_{r}(s)}}{\Delta \; {i_{r}(s)}} \right|_{{\Delta \; i_{r}} = 0} = \frac{m^{2}I_{r}}{4s\; C_{{dc},r}}}}} & (24) \end{matrix}$

Turning now to FIG. 7, there is shown a small signal control block diagram 700 of the power flow control apparatus 402. In the control block diagram 700, a proportional-plus-integral (PI) controller H_(r) is used to regulate the output voltage v_(r) at the reference voltage V_(ref,r) by altering the angle θ_(r) in each compensator unit. In one embodiment, the closed-loop transfer function of each compensator is

$\begin{matrix} \begin{matrix} {{F_{r}(s)} = \frac{\Delta \; {v_{r}(s)}}{{\Delta V}_{{ref},r}(s)}} \\ {= \frac{{H_{r}(s)}K_{m}{D_{\theta}(s)}}{1 + {{H_{r}(s)}K_{m}{D_{\theta}(s)}}}} \\ {= \frac{m^{2}I_{r}K_{p\;}{K_{m}\left( {s + \frac{K_{i}}{K_{p}}} \right)}}{4{C_{{dc},r}\left( {s^{2} + {\frac{m^{2}I_{r}K_{p}K_{m}}{4C_{{dc},r}}s} + \frac{m^{2}I_{r}K_{i}K_{m}}{4C_{{dc},r}}} \right)}}} \end{matrix} & (25) \end{matrix}$

where H_(r)

${(s) = \left( {K_{p} + {K_{i}\frac{1}{s}}} \right)},$

in which K_(P) and K_(i) are the proportional gain and the integral gain of H_(r), respectively, and K_(m)=π/180 is the gain of the modulator. In this embodiment, the closed-loop poles of F_(r)(s) are all lying in the left-half s plane. This indicates the stability of the response of the power flow control apparatus.

The transient response of the power flow control apparatus is further investigated by considering the maximum overshoot M_(p) and the settling time t_(s).

To investigate the response of the power flow control apparatus, a step function is applied to the closed-loop transfer function F_(r)(s) to calculate M_(p) and t_(s). FIG. 8A shows the relationship of the maximum overshoot M_(p) against the integral gain K_(i) and the proportional gain K_(p) of the power flow control apparatus under different values of current I_(r) whereas FIG. 8B shows the relationship of the settling time t_(s) against the integral gain K_(i) and the proportional gain K_(p) of the power flow control apparatus under different values of current I_(r). In this example, the settling time is based on the 2% criterion. By substituting equation (25) into equation (19), it can be shown that

Δi(s)=−G(s)F(s)Δv _(ref)(s)  (26)

where

${F(s)} = \begin{bmatrix} {F_{1}(s)} & 0 & \ldots & 0 \\ 0 & {F_{2}(s)} & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & 0 & \ldots & {F_{N}(s)} \end{bmatrix}$

and ΔV_(ref)(s)=[ΔV_(ref,1)(s)ΔV_(ref,2)(s) . . . ΔV_(ref,N)(s)]^(T).

An embodiment of a design procedure of the power flow control apparatus 402 is described below.

Wither reference to FIGS. 4-6, preferably, L_(q,r) and C_(q,r) are designed by considering the maximum voltage drop across v_(Lq,r), and the maximum current ripple ĩ_(r) respectively. More particularly, L_(q,r) and C_(q,r) are designed based on the criterion below:

$\begin{matrix} {{L_{q,r} < \frac{v_{{Lq},r}}{2\pi \; f\; I_{r}}},{C_{q,r} > {\frac{1}{2\pi \; {f_{s}\left( {{2\pi \; f_{s}L_{q,r}} - \frac{4V_{dc}}{\pi \; {\overset{\sim}{i}}_{r}}} \right)}}.}}} & (27) \end{matrix}$

With further reference to equation (21), C_(dc,r) is preferably chosen to be

$\begin{matrix} {C_{{dc},r} = {\frac{m}{8\omega}\frac{I_{r}}{{\hat{v}}_{{dc},r}}}} & (28) \end{matrix}$

where {circumflex over (v)}_(dc,r) is the peak value of the ripple voltage on C_(dc,r). As for the design of transformer T_(r), preferably, a core geometry approach method is used.

FIGS. 9-11 showed the experimental verification of the power flow control apparatus of the present invention. In particular, a testing setup with three parallel-connected SSSC using daisy-chained transformers has been built and evaluated. In this experiment, with further reference to FIG. 4, v_(A) and v_(B) are chosen to be 225V, 50 Hz, with phase difference of 5°. The inductance L and resistance R are chosen to be 20 mH and 0.2Ω respectively. The transformers T₁, T₂ and T₃ have the magnetizing inductance of 162 mH, 152 mH and 160 mH respectively. In FIGS. 9-11, v₁, v₂, v₃ are compensator voltages whereas i₁, i₂, i₃ are compensator currents.

FIG. 9 shows the steady-state voltage waveforms 900 and the steady-state current waveforms 902 of the compensator units 406 when the power transferring from v_(A) to v_(B) is 965 W. FIG. 10 shows the steady-state voltage waveforms 1000 and the steady-state current waveforms 1002 of the compensator units 406 when the power transferring from v_(B) to v_(A) is 595 W. It is observed that in this example the voltage and current waveforms of the three compensator units are similar. FIG. 11 shows the transient voltage waveforms 1100 and transient current waveforms 1102 of the compensator units 406 when the power flow control apparatus is changed from the operation of FIG. 9 to the operation of FIG. 10. It is observed that the transient response of the compensator currents and voltages gradually decays and a stable steady state response is obtained. More importantly, these results are in agreement with the modelling and analysis of the power flow control apparatus described above.

In summary, an embodiment of a power flow control apparatus based on connecting multiple units of SSSC in parallel to form a modular-based multi-parallel-connected SSSC (MSSSC) architecture has been described. Particularly, the power flow control apparatus also comprises a daisy chained transformer structure. The presented power flow control apparatus of this invention may be advantageous in that it enables the concept of “plug-and produce”, i.e. the apparatus has a high degree of modularity, scalability, adaptability and autonomic behaviour. In a particular embodiment, each SSSC compensator unit in the MSSSC is coupled to one another through a coupling transformer so as to make each unit share the transmission current equally.

In some applications, such modular approach allows consumers on the demand side to flexibly optimize and manage the power flow in the microgrid and manufacturers to produce standardized low-power units for distributed power system. Moreover, in some embodiments, each SSSC unit is fully autonomic, as it will self-regulate the injection (compensating) voltage after receiving the command from the advanced metering infrastructure. Advantageously, in situations when one of the SSSC units malfunctions, the other SSSC units will not be affected and the power flow control apparatus can still function with a lower power transfer.

It will be appreciated by persons skilled in the art that numerous variations and/or modifications may be made to the invention as shown in the specific embodiments without departing from the spirit or scope of the invention as broadly described. The present embodiments are, therefore, to be considered in all respects as illustrative and not restrictive.

Any reference to prior art contained herein is not to be taken as an admission that the information is common general knowledge, unless otherwise indicated. 

1. A power flow control apparatus comprising: a current distribution circuit arranged to distribute an input current into a plurality of branches such that the input current is distributed into a plurality of individual branch currents; wherein each of the plurality of branches includes an inductive arrangement arranged to form an inductive coupling with an associated inductive arrangement of at least one other associated branch; and a plurality of compensator units in electrical communication with the plurality of branches, wherein each compensator unit is arranged to deliver a branch compensating voltage relative to the branch current.
 2. A power flow control apparatus in accordance with claim 1, wherein the compensator units are disposed downstream of the current distribution circuit.
 3. A power flow control apparatus in accordance with claim 2, wherein each compensator unit is arranged to regulate its branch compensating voltage.
 4. A power flow control apparatus in accordance with claim 2, wherein each compensator unit is further arranged to automatically regulate its branch compensating voltage in response to a command.
 5. A power flow control apparatus in accordance with claim 4, wherein the power flow control apparatus is further arranged to provide an output voltage based on the branch compensating voltages of the compensator units.
 6. A power flow control apparatus in accordance with claim 1, wherein the input current is a current on a transmission line having at least one current source.
 7. A power flow control apparatus in accordance with claim 1, wherein the current distribution circuit distributes the input current into the plurality of individual branch currents based on at least one predetermined ratio of the current distribution circuit.
 8. A power flow control apparatus in accordance with claim 7, wherein the inductive arrangement of each branch of the current distribution circuit comprises a first coil and a second coil.
 9. A power flow control apparatus in accordance with claim 8, wherein the at least one predetermined ratio of the current distribution circuit is associated with a number of turns of the first coil and a number of turns of the second coil.
 10. A power flow control apparatus in accordance with claim 8, wherein the first coil of each branch is inductively coupled with the second coil of an adjacent branch.
 11. A power flow control apparatus in accordance with claim 8, wherein the first coil of each branch and the second coil of an adjacent branch together defines a transformer unit.
 12. A power flow control apparatus in accordance with claim 11, wherein the first coils are primary coils of the transformer units and the second coils are secondary coils of the transformer units.
 13. A power flow control apparatus in accordance with claim 12, wherein the primary coil and the secondary coil of each transformer unit of the current distribution circuit are disposed in adjacent branches.
 14. A power flow control apparatus in accordance with claim 12, wherein the primary coil of each transformer unit of the current distribution circuit is disposed on the same branch with the secondary coil of an adjacent transformer unit of the current distribution circuit.
 15. A power flow control apparatus in accordance with claim 12, wherein the primary coils each comprises a first number of turns of coils and the secondary coils each comprises a second number of turns of coils.
 16. A power flow control apparatus in accordance with claim 15, wherein each transformer unit of the current distribution circuit has a turn ratio defined by the first number of turns of coils of the primary coil and the second number of turns of coils of the secondary coil.
 17. A power flow control apparatus in accordance with claim 16, wherein the at least one predetermined ratio of the current distribution circuit is determined by at least one of the turn ratio of at least one of the transformer units.
 18. A power flow control apparatus in accordance with claim 1, wherein a summation of the individual branch current of each of the plurality of individual branches of the current distribution circuit is substantially equal to the input current.
 19. A power flow control apparatus in accordance with claim 1, wherein the compensator units are static synchronous series compensators (SSSC) each comprising a voltage source converter.
 20. A power flow control apparatus in accordance with claim 19, wherein each static synchronous series compensator is arranged to provide the branch compensating voltage.
 21. A power flow control apparatus in accordance with claim 20, wherein each static synchronous series compensator is further arranged to regulate its branch compensating voltage.
 22. A power flow control apparatus in accordance with claim 21, wherein each static synchronous series compensator is further arranged to automatically regulate its branch compensating voltage in response to a command.
 23. A power flow control apparatus in accordance with claim 22, wherein the power flow control apparatus is arranged to provide an output voltage based on the branch compensating voltages of the static synchronous series compensators.
 24. A power flow control apparatus in accordance with claim 11, wherein the number of transformer units equals the number of branches.
 25. A power flow control apparatus in accordance with claim 1, wherein the number of compensator units equals the number of branches.
 26. A power flow control apparatus in accordance with claim 11, wherein the number of compensator units equals the number of transformer units.
 27. A power flow control apparatus in accordance with claim 11, wherein the transformer units are connected in a daisy-chained manner. 